Vector spaces and linear maps
... Definition 2 A vector space V over F is called finite dimensional if there is a finite subset {v1 , . . . , vn } of elements of V such that Span{v1 , . . . , vn } = V . Notice that F n is finite dimensional (the standard basis spans), while C([0, 1]) is not. Just as for subspaces of Rn : Definition ...
... Definition 2 A vector space V over F is called finite dimensional if there is a finite subset {v1 , . . . , vn } of elements of V such that Span{v1 , . . . , vn } = V . Notice that F n is finite dimensional (the standard basis spans), while C([0, 1]) is not. Just as for subspaces of Rn : Definition ...
MS Word
... 2. Consider the group < J+7, +7 >. Find 3-1 and 5-1. The identity for this group is 0. (3 +7 4) = 0, so 3-1 = 4. Similarly, (5 +7 2) = 0, so 5-1 = 2. 3. Consider the group. Find the identity for this group, as well as 2-1. The
identity for this group is also 0, thus (2 +4 2) = 0, so 2-1 = ...
... 2. Consider the group < J+7, +7 >. Find 3-1 and 5-1. The identity for this group is 0. (3 +7 4) = 0, so 3-1 = 4. Similarly, (5 +7 2) = 0, so 5-1 = 2. 3. Consider the group
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... A function T : V W is called one - to - one if the preimage of every w in the range consists of a single vector. T is one - to - one iff for all u and v inV, T (u) T ( v) implies that u v. ...
... A function T : V W is called one - to - one if the preimage of every w in the range consists of a single vector. T is one - to - one iff for all u and v inV, T (u) T ( v) implies that u v. ...
2.7 Banach spaces
... right side of these equations, we use the vector space operations in Y ). Then f + g and αf are also in C(X, Y ), so C(X, Y ) is a vector space over K. For f ∈ C(X, Y ), the function x 7→ kf (x)k0 is a continuous function on the compact set X, since it is the composition of f : X → Y and k k0 : Y → ...
... right side of these equations, we use the vector space operations in Y ). Then f + g and αf are also in C(X, Y ), so C(X, Y ) is a vector space over K. For f ∈ C(X, Y ), the function x 7→ kf (x)k0 is a continuous function on the compact set X, since it is the composition of f : X → Y and k k0 : Y → ...
4.3 Linear Combinations and Spanning Sets
... called the standard spanning set of R2 . There are others, as the next example suggests. Example 279 Show that S = fu1 ; u2 g where u1 = (1; 1) and u2 = (0; 1) spans R2 Here, we have to show that any element of R2 can be expressed as a linear combination of vectors of S. A typical element of R2 is o ...
... called the standard spanning set of R2 . There are others, as the next example suggests. Example 279 Show that S = fu1 ; u2 g where u1 = (1; 1) and u2 = (0; 1) spans R2 Here, we have to show that any element of R2 can be expressed as a linear combination of vectors of S. A typical element of R2 is o ...
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (""scaled"") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.